(Artificial) Neural Networks (ANN)
By Iljeok Kim and Sooyoung Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
Table of Contents
1. Recall Supervised Learning SetupĀ¶
Perceptron
XOR Problem
- Minsky-Papert Controversy on XOR
- not linearly separable
- limitation of perceptron
| $x_1$ | $x_2$ | $x_1$ XOR $x_2$ |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
2. From Perceptron to Multi-Layer Perceptron (MLP)Ā¶
2.1. Perceptron for $h_{\omega}(x)$Ā¶
Neurons compute the weighted sum of their inputs
A neuron is activated or fired when the sum $a$ is positive
$$
\begin{align*}
a &= \omega_0 + \omega_1 x_1 + \omega_2 x_2 \\ \\
\hat{y} &= g(a) =
\begin{cases}
1 & a > 0\\
0 & \text{otherwise}
\end{cases}
\end{align*}
$$
- A step function is not differentiable
- One layer is often not enough
- One hyperplane
Differentiable activation function
In a compact representation
Multi-layer perceptron
2.3. Another Perspective: ANN as Kernel LearningĀ¶
We can represent this āneuronā as follows:
The main weakness of linear predictors is their lack of capacity. For classiļ¬cation, the populations have to be linearly separable.
The XOR example can be solved by pre-processing the data to make the two populations linearly separable.
Universal function approximator Universal function classifier
Parameterized
Example: Linear Classifier
- Perceptron tries to separate the two classes of data by dividing them with a line
Example: Neural Networks
- The hidden layer learns a representation so that the data is linearly separable
colah's blog
- Neural Networks, Manifolds, and Topology
- http://colah.github.io/posts/2014-03-NN-Manifolds-Topology/
InĀ [1]:
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
import time
%matplotlib inline
InĀ [2]:
#training data gerneration
m = 1000
x1 = 8*np.random.rand(m, 1)
x2 = 7*np.random.rand(m, 1) - 4
g = 0.8*x1 + x2 - 3
C1 = np.where(g >= 0)[0]
C0 = np.where(g < 0)[0]
N = C1.shape[0]
M = C0.shape[0]
m = N + M
X1 = np.hstack([np.ones([N,1]), x1[C1], x2[C1]])
X0 = np.hstack([np.ones([M,1]), x1[C0], x2[C0]])
train_X = np.vstack([X1, X0])
train_y = np.vstack([np.ones([N,1]), -np.ones([M,1])])
train_X = np.asmatrix(train_X)
train_y = np.asmatrix(train_y)
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.legend(loc = 1, fontsize = 15)
plt.xlabel(r'$x_1$', fontsize = 15)
plt.ylabel(r'$x_2$', fontsize = 15)
plt.show()
InĀ [3]:
train_y = np.vstack([np.ones([N,1]), np.zeros([M,1])])
train_y = np.asmatrix(train_y)
InĀ [4]:
import tensorflow as tf
LR = 0.05
n_iter = 10000
x = tf.placeholder(tf.float32, [None, 3])
y = tf.placeholder(tf.float32, [None, 1])
w = tf.Variable(tf.random_normal([3,1]))
# y_predė„¼ źµ¬ķģ¬ģ¼ ķØ
y_pred = tf.matmul(?,?)
loss = tf.nn.sigmoid_cross_entropy_with_logits(logits = y_pred, labels = y)
loss = tf.reduce_mean(loss)
optm = tf.train.GradientDescentOptimizer(LR).minimize(loss)
init = tf.global_variables_initializer()
InĀ [5]:
# xģ yģ ķģµķ ė°ģ“ķ°ė„¼ ģ£¼ģ
ķ“ģ¼ķØ
with tf.Session() as sess:
sess.run(init)
for epoch in range(n_iter):
sess.run(optm, feed_dict = {x: ?, y: ?})
w_hat = sess.run(w)
InĀ [6]:
x1p = np.arange(0, 8, 0.01).reshape(-1, 1)
x2p = - w_hat[1,0]/w_hat[2,0]*x1p - w_hat[0,0]/w_hat[2,0]
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'g', linewidth = 3, label = '')
plt.xlim([0, 8])
plt.xlabel('$x_1$', fontsize = 15)
plt.ylabel('$x_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 12)
plt.show()
3.2. Logistic Regression in a Form of Neural NetworkĀ¶
$$y = \sigma \,(\omega_0 + \omega_1 x_1 + \omega_2 x_2)$$
InĀ [7]:
# define input and output size
n_input = 3
n_output = 1
InĀ [8]:
# define weights as a dictionary
weights = {
'output' : tf.Variable(tf.random_normal([n_input, n_output], stddev = 0.1))
}
InĀ [9]:
# define placeholders for train_x and train_y
x = tf.placeholder(tf.float32, [None, ?])
y = tf.placeholder(tf.float32, [None, ?])
InĀ [10]:
# define network architecture
def build_model(x, weights):
output = tf.matmul(x, weights['output'])
return output
InĀ [11]:
# define loss
pred = build_model(x, weights)
loss = tf.nn.sigmoid_cross_entropy_with_logits(logits = pred, labels = y)
loss = tf.reduce_mean(loss)
InĀ [12]:
LR = 0.05
optm = tf.train.GradientDescentOptimizer(LR).minimize(loss)
InĀ [13]:
n_batch = 50 # Batch size
n_iter = 10000 # Learning iteration
n_prt = 250 # Print cycle
sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
# training or learning
loss_record = []
for epoch in range(n_iter):
sess.run(optm, feed_dict = {x: train_X, y: train_y})
if epoch % n_prt == 0:
loss_record.append(sess.run(loss, feed_dict = {x: train_X, y: train_y}))
w_hat = sess.run(weights['output'])
InĀ [14]:
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record))*n_prt, loss_record)
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.show()
InĀ [15]:
x1p = np.arange(0, 8, 0.01).reshape(-1, 1)
x2p = - w_hat[1,0]/w_hat[2,0]*x1p - w_hat[0,0]/w_hat[2,0]
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'g', linewidth = 3, label = '')
plt.xlim([0, 8])
plt.xlabel('$x_1$', fontsize = 15)
plt.ylabel('$x_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 12)
plt.show()
Weights and Bias
- In a neural network, weights and biases are typically separated.
$$ \begin{align*} y_j &= \left(\sum\limits_i \omega_{ij}x_i\right) + b_j\\ y &= \omega^T \mathcal{x} + \mathcal{b} \end{align*} $$
InĀ [16]:
n_input = 2
n_output = 1
InĀ [17]:
train_X = train_X[:,1:3]
InĀ [18]:
# define network
def build_model(x, weights, biases):
output = tf.add(tf.matmul(x, weights['output']), biases['output'])
return output
InĀ [19]:
weights = {
'output' : tf.Variable(tf.random_normal([n_input, n_output], stddev = 0.1))
}
biases = {
'output' : tf.Variable(tf.random_normal([n_output], stddev = 0.1))
}
x = tf.placeholder(tf.float32, [None, n_input])
y = tf.placeholder(tf.float32, [None, n_output])
# ģģø”ź°ź³¼ ģ ėµģ ģ°Øģ“ė„¼ ė¹źµķ“ģ¼ķØ
pred = build_model(x, weights, biases)
loss = tf.nn.sigmoid_cross_entropy_with_logits(logits=?, labels=?)
loss = tf.reduce_mean(loss)
LR = 0.05
optm = tf.train.GradientDescentOptimizer(LR).minimize(loss)
sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
n_batch = 50
n_iter = 15000
n_prt = 250
loss_record = []
for epoch in range(n_iter):
sess.run(optm, feed_dict = {x: train_X, y: train_y})
if epoch % n_prt == 0:
loss_record.append(sess.run(loss, feed_dict = {x: train_X, y: train_y}))
w_hat = sess.run(weights['output'])
b_hat = sess.run(biases['output'])
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record))*n_prt, loss_record)
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.show()
InĀ [20]:
x1p = np.arange(0, 8, 0.01).reshape(-1, 1)
x2p = - w_hat[0,0]/w_hat[1,0]*x1p - b_hat[0]/w_hat[1,0]
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'g', linewidth = 3, label = '')
plt.xlim([0, 8])
plt.xlabel('$x_1$', fontsize = 15)
plt.ylabel('$x_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 12)
plt.show()
One-hot Encoding
- One-hot encoding is a conventional practice for a multi-class classification
$$y^{(i)} \in \{1,0\} \quad \implies \quad y^{(i)} \in \{[0,1],[1,0]\}$$
tf.nn.sigmoid_cross_entropy_with_logits$\rightarrow$tf.nn.softmax_cross_entropy_with_logits
InĀ [21]:
from sklearn.preprocessing import OneHotEncoder
ohe = OneHotEncoder(handle_unknown='ignore')
train_y = ohe.fit_transform(train_y).toarray()
print(train_y)
InĀ [22]:
# output nodeģ ź°ģź° ė³ķØ
n_input = 2
n_output = ?
InĀ [23]:
weights = {
'output' : tf.Variable(tf.random_normal([n_input, n_output], stddev = 0.1))
}
biases = {
'output' : tf.Variable(tf.random_normal([n_output], stddev = 0.1))
}
x = tf.placeholder(tf.float32, [None, n_input])
y = tf.placeholder(tf.float32, [None, n_output])
pred = build_model(x, weights, biases)
loss = tf.nn.softmax_cross_entropy_with_logits(logits=pred, labels=y)
loss = tf.reduce_mean(loss)
LR = 0.05
optm = tf.train.GradientDescentOptimizer(LR).minimize(loss)
sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
n_batch = 50
n_iter = 10000
n_prt = 250
loss_record = []
for epoch in range(n_iter):
sess.run(optm, feed_dict = {x: train_X, y: train_y})
if epoch % n_prt == 0:
loss_record.append(sess.run(loss, feed_dict = {x: train_X, y: train_y}))
w_hat = sess.run(weights['output'])
b_hat = sess.run(biases['output'])
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record))*n_prt, loss_record)
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.show()
InĀ [24]:
print(w_hat)
x1p = np.arange(0, 8, 0.01).reshape(-1, 1)
x2p = - w_hat[0,0]/w_hat[1,0]*x1p - b_hat[0]/w_hat[1,0]
x3p = - w_hat[0,1]/w_hat[1,1]*x1p - b_hat[1]/w_hat[1,1]
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'k', linewidth = 3, label = '')
plt.plot(x1p, x3p, 'g', linewidth = 3, label = '')
plt.xlim([0, 8])
plt.xlabel('$x_1$', fontsize = 15)
plt.ylabel('$x_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 12)
plt.show()
InĀ [25]:
# training data gerneration
m = 1000
x1 = 10*np.random.rand(m, 1) - 5
x2 = 8*np.random.rand(m, 1) - 4
g = - 0.5*(x1-1)**2 + 2*x2 + 5
C1 = np.where(g >= 0)[0]
C0 = np.where(g < 0)[0]
N = C1.shape[0]
M = C0.shape[0]
m = N + M
X1 = np.hstack([x1[C1], x2[C1]])
X0 = np.hstack([x1[C0], x2[C0]])
train_X = np.vstack([X1, X0])
train_X = np.asmatrix(train_X)
train_y = np.vstack([np.ones([N,1]), np.zeros([M,1])])
ohe = OneHotEncoder(handle_unknown='ignore')
train_y = ohe.fit_transform(train_y).toarray()
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.legend(loc = 1, fontsize = 15)
plt.xlabel(r'$x_1$', fontsize = 15)
plt.ylabel(r'$x_2$', fontsize = 15)
plt.xlim([-5, 5])
plt.ylim([-4, 4])
plt.show()
InĀ [26]:
# nodeģ ź°ģ
n_input = ?
n_hidden = ?
n_output = ?
InĀ [27]:
# weights (ķģ“ķ)ģ ź°ģ
weights = {
'hidden' : tf.Variable(tf.random_normal([n_input, ?], stddev = 0.1)),
'output' : tf.Variable(tf.random_normal([?, n_output], stddev = 0.1))
}
biases = {
'hidden' : tf.Variable(tf.random_normal([n_hidden], stddev = 0.1)),
'output' : tf.Variable(tf.random_normal([n_output], stddev = 0.1))
}
x = tf.placeholder(tf.float32, [None, n_input])
y = tf.placeholder(tf.float32, [None, n_output])
InĀ [28]:
def build_model(x, weights, biases):
hidden = tf.add(tf.matmul(x, weights['hidden']), biases['hidden'])
hidden = tf.nn.sigmoid(hidden)
output = tf.add(tf.matmul(hidden, weights['output']), biases['output'])
return output
InĀ [29]:
# ģģø”ź°ź³¼ ģ ėµģ ė¹źµ
pred = build_model(x, weights, biases)
loss = tf.nn.softmax_cross_entropy_with_logits(logits = ?, labels = ?)
loss = tf.reduce_mean(loss)
LR = 0.01
optm = tf.train.GradientDescentOptimizer(LR).minimize(loss)
sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
n_batch = 50
n_iter = 30000
n_prt = 250
# ķģµķ ė°ģ“ķ° ģ£¼ģ
loss_record = []
for epoch in range(n_iter):
sess.run(optm, feed_dict = {x: ?, y: ?})
if epoch % n_prt == 0:
loss_record.append(sess.run(loss, feed_dict = {x: ?, y: ?}))
w_hat = sess.run(weights)
b_hat = sess.run(biases)
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record))*n_prt, loss_record)
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.show()
InĀ [30]:
H = train_X*w_hat['hidden'] + b_hat['hidden']
H = 1/(1 + np.exp(-H))
InĀ [31]:
plt.figure(figsize=(10, 8))
plt.plot(H[0:N,0], H[0:N,1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(H[N:m,0], H[N:m,1], 'bo', alpha = 0.4, label = 'C0')
plt.xlabel('$z_1$', fontsize = 15)
plt.ylabel('$z_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 15)
plt.axis('equal')
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.show()
InĀ [32]:
x1p = np.arange(0, 1, 0.01).reshape(-1, 1)
x2p = - w_hat['output'][0,0]/w_hat['output'][1,0]*x1p - b_hat['output'][0]/w_hat['output'][1,0]
x3p = - w_hat['output'][0,1]/w_hat['output'][1,1]*x1p - b_hat['output'][1]/w_hat['output'][1,1]
plt.figure(figsize=(10, 8))
plt.plot(H[0:N,0], H[0:N,1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(H[N:m,0], H[N:m,1], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'k', linewidth = 3, label = '')
plt.plot(x1p, x3p, 'g', linewidth = 3, label = '')
plt.xlabel('$z_1$', fontsize = 15)
plt.ylabel('$z_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 15)
plt.axis('equal')
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.show()
InĀ [33]:
x1p = np.arange(-5, 5, 0.01).reshape(-1, 1)
x2p = - w_hat['hidden'][0,0]/w_hat['hidden'][1,0]*x1p - b_hat['hidden'][0]/w_hat['hidden'][1,0]
x3p = - w_hat['hidden'][0,1]/w_hat['hidden'][1,1]*x1p - b_hat['hidden'][1]/w_hat['hidden'][1,1]
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'k', linewidth = 3, label = '')
plt.plot(x1p, x3p, 'g', linewidth = 3, label = '')
plt.xlabel('$x_1$', fontsize = 15)
plt.ylabel('$x_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 15)
plt.axis('equal')
plt.xlim([-5, 5])
plt.ylim([-4, 4])
plt.show()
4.2. Multi-NeuronsĀ¶
InĀ [34]:
# training data gerneration
m = 1000
x1 = 10*np.random.rand(m, 1) - 5
x2 = 8*np.random.rand(m, 1) - 4
g = - 0.5*(x1*x2-1)**2 + 2*x2 + 5
C1 = np.where(g >= 0)[0]
C0 = np.where(g < 0)[0]
N = C1.shape[0]
M = C0.shape[0]
m = N + M
X1 = np.hstack([x1[C1], x2[C1]])
X0 = np.hstack([x1[C0], x2[C0]])
train_X = np.vstack([X1, X0])
train_X = np.asmatrix(train_X)
train_y = np.vstack([np.ones([N,1]), np.zeros([M,1])])
ohe = OneHotEncoder(handle_unknown='ignore')
train_y = ohe.fit_transform(train_y).toarray()
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.legend(loc = 1, fontsize = 15)
plt.xlabel(r'$x_1$', fontsize = 15)
plt.ylabel(r'$x_2$', fontsize = 15)
plt.xlim([-5, 5])
plt.ylim([-4, 4])
plt.show()
InĀ [35]:
n_input = ?
n_hidden = ?
n_output = ?
InĀ [36]:
def build_model(x, weights, biases):
hidden = tf.add(tf.matmul(x, weights['hidden']), biases['hidden'])
hidden = tf.nn.sigmoid(hidden)
output = tf.add(tf.matmul(hidden, weights['output']), biases['output'])
return output
InĀ [37]:
weights = {
'hidden' : tf.Variable(tf.random_normal([?, ?], stddev = 0.1)),
'output' : tf.Variable(tf.random_normal([?, ?], stddev = 0.1))
}
biases = {
'hidden' : tf.Variable(tf.random_normal([?], stddev = 0.1)),
'output' : tf.Variable(tf.random_normal([?], stddev = 0.1))
}
x = tf.placeholder(tf.float32, [None, ?])
y = tf.placeholder(tf.float32, [None, ?])
InĀ [38]:
pred = build_model(?, ?, ?)
loss = tf.nn.softmax_cross_entropy_with_logits(logits = ?, labels = ?)
loss = tf.reduce_mean(?)
LR = 0.01
optm = tf.train.GradientDescentOptimizer(LR).minimize(loss)
sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
n_batch = 50
n_iter = 40000
n_prt = 250
# Training cycle
loss_record = []
for epoch in range(n_iter):
sess.run(optm, feed_dict = {x: ?, y: ?})
if epoch % n_prt == 0:
loss_record.append(sess.run(loss, feed_dict = {x: ?, y: ?}))
w_hat = sess.run(weights)
b_hat = sess.run(biases)
# plots
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record))*n_prt, loss_record)
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.show()
InĀ [39]:
x1p = np.arange(-5, 5, 0.01).reshape(-1, 1)
x2p = - w_hat['hidden'][0,0]/w_hat['hidden'][1,0]*x1p - b_hat['hidden'][0]/w_hat['hidden'][1,0]
x3p = - w_hat['hidden'][0,1]/w_hat['hidden'][1,1]*x1p - b_hat['hidden'][1]/w_hat['hidden'][1,1]
x4p = - w_hat['hidden'][0,2]/w_hat['hidden'][1,2]*x1p - b_hat['hidden'][2]/w_hat['hidden'][1,2]
x5p = - w_hat['hidden'][0,3]/w_hat['hidden'][1,3]*x1p - b_hat['hidden'][3]/w_hat['hidden'][1,3]
plt.figure(figsize=(10, 8))
plt.plot(x1[C1], x2[C1], 'ro', alpha = 0.4, label = 'C1')
plt.plot(x1[C0], x2[C0], 'bo', alpha = 0.4, label = 'C0')
plt.plot(x1p, x2p, 'k', linewidth = 3, label = '')
plt.plot(x1p, x3p, 'g', linewidth = 3, label = '')
plt.plot(x1p, x4p, 'm', linewidth = 3, label = '')
plt.plot(x1p, x5p, 'c', linewidth = 3, label = '')
plt.xlabel('$x_1$', fontsize = 15)
plt.xlabel('$x_1$', fontsize = 15)
plt.ylabel('$x_2$', fontsize = 15)
plt.legend(loc = 1, fontsize = 15)
plt.axis('equal')
plt.xlim([-5, 5])
plt.ylim([-4, 4])
plt.show()
5. Artificial Neural NetworksĀ¶
- Complex/Nonlinear universal function approximator
- Linearly connected networks
- Simple nonlinear neurons
- Hidden layers
- Autonomous feature learning
5.1. Recursive AlgorithmĀ¶
One of the central ideas of computer science
Depends on solutions to smaller instances of the same problem ( = subproblem)
Function to call itself (it is impossible in the real world)
- Factorial example
$$n ! = n \cdot (n-1) \cdots 2 \cdot 1$$
InĀ [40]:
n = 5
m = 1
for i in range(n):
m = m*(i+1)
print(m)
InĀ [41]:
def fac(n):
if n == 1:
return 1
else:
return n*fac(n-1)
InĀ [42]:
# recursive
fac(5)
Out[42]:
5.2. Dynamic ProgrammingĀ¶
Dynamic Programming: general, powerful algorithm design technique
Fibonacci numbers:
InĀ [43]:
# naive Fibonacci
# write a function called fib
InĀ [44]:
fib(10)
Out[44]:
InĀ [45]:
# Memorized DP Fibonacci
def mfib(n):
global memo
if memo[n-1] != 0:
return memo[n-1]
elif n <= 2:
memo[n-1] = 1
return memo[n-1]
else:
memo[n-1] = mfib(n-1) + mfib(n-2)
return memo[n-1]
InĀ [46]:
import numpy as np
n = 10
memo = np.zeros(n)
mfib(n)
Out[46]:
InĀ [47]:
n = 30
%timeit fib(30)
InĀ [48]:
memo = np.zeros(n)
%timeit mfib(30)
5.3. Training Neural NetworksĀ¶
$=$ Learning or estimating weights and biases of multi-layer perceptron from training data
5.3.1. OptimizationĀ¶
3 key components
- objective function $f(\cdot)$
- decision variable or unknown $\omega$
- constraints $g(\cdot)$
In mathematical expression
$$\begin{align*}
\min_{\omega} \quad &f(\omega)
\end{align*}
$$
5.3.2. Loss FunctionĀ¶
- Measures error between target values and predictions
$$ \min_{\omega} \sum_{i=1}^{m}\ell\left( h_{\omega}\left(x^{(i)}\right),y^{(i)}\right)$$
- Example
- Squared loss (for regression):
$$ \frac{1}{m} \sum_{i=1}^{m} \left(h_{\omega}\left(x^{(i)}\right) - y^{(i)}\right)^2 $$
- Cross entropy (for classification): $$ -\frac{1}{m}\sum_{i=1}^{m}y^{(i)}\log\left(h_{\omega}\left(x^{(i)}\right)\right) + \left(1-y^{(i)}\right)\log\left(1-h_{\omega}\left(x^{(i)}\right)\right)$$
- Squared loss (for regression):
$$ \frac{1}{m} \sum_{i=1}^{m} \left(h_{\omega}\left(x^{(i)}\right) - y^{(i)}\right)^2 $$
5.3.3. LearningĀ¶
Learning weights and biases from data using gradient descent
- $\frac{\partial \ell}{\partial \omega}$: too many computations are required for all $\omega$
- Structural constraints of NN:
- Composition of functions
- Chain rule
- Dynamic programming
Backpropagation
- Forward propagation
- the initial information propagates up to the hidden units at each layer and finally produces output
- Backpropagation
- allows the information from the cost to flow backwards through the network in order to compute the gradients
Chain Rule
Computing the derivative of the composition of functions
$\space f(g(x))' = f'(g(x))g'(x)$
$\space {dz \over dx} = {dz \over dy} \bullet {dy \over dx}$
$\space {dz \over dw} = ({dz \over dy} \bullet {dy \over dx}) \bullet {dx \over dw}$
$\space {dz \over du} = ({dz \over dy} \bullet {dy \over dx} \bullet {dx \over dw}) \bullet {dw \over du}$
Backpropagation
Update weights recursively with memory
Optimization procedure
- It is not easy to numerically compute gradients in network in general.
- The good news: people have already done all the "hardwork" of developing numerical solvers (or libraries)
- There are a wide range of tools: TensorFlow
Summary
- Learning weights and biases from data using gradient descent
6. ANN with MNISTĀ¶
6.1. What's an MNIST?Ā¶
From Wikipedia
- The MNIST database (Mixed National Institute of Standards and Technology database) is a large database of handwritten digits that is commonly used for training various image processing systems. The database is also widely used for training and testing in the field of machine learning. It was created by "re-mixing" the samples from NIST's original datasets. The creators felt that since NIST's training dataset was taken from American Census Bureau employees, while the testing dataset was taken from American high school students, NIST's complete dataset was too hard.
- MNIST (Mixed National Institute of Standards and Technology database) database
- Handwritten digit database
- $28 \times 28$ gray scaled image
- Flattened matrix into a vector of $28 \times 28 = 784$
More here
We will be using MNIST to create a Multinomial Classifier that can detect if the MNIST image shown is a member of class 0,1,2,3,4,5,6,7,8 or 9. Susinctly, we're teaching a computer to recognize hand written digets.
InĀ [Ā ]:
# import os
# os.environ["CUDA_DEVICE_ORDER"]="PCI_BUS_ID"
# os.environ["CUDA_VISIBLE_DEVICES"]="0"
InĀ [49]:
# Import Library
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
%matplotlib inline
Let's download and load the dataset.
InĀ [50]:
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("MNIST_data/", one_hot=True)
InĀ [51]:
print ("The training data set is:\n")
print (mnist.train.images.shape)
print (mnist.train.labels.shape)
InĀ [52]:
print ("The test data set is:")
print (mnist.test.images.shape)
print (mnist.test.labels.shape)
Display a few random samples from it:
InĀ [53]:
mnist.train.images[5]
Out[53]:
InĀ [54]:
# well, that's not a picture (or image), it's an array.
mnist.train.images[5].shape
Out[54]:
You might think the training set is made up of 28 $\times$28 grayscale images of handwritten digits. No !!!
The thing is, the image has been flattened. These are 28x28 images that have been flattened into a 1D array. Let's reshape one.
InĀ [55]:
img = np.reshape(mnist.train.images[5], [28,28])
InĀ [56]:
img = mnist.train.images[5].reshape([28,28])
InĀ [57]:
# So now we have a 28x28 matrix, where each element is an intensity level from 0 to 1.
img.shape
Out[57]:
Let's visualize what some of these images and their corresponding training labels look like.
InĀ [58]:
plt.figure(figsize = (6,6))
plt.imshow(img, 'gray')
plt.xticks([])
plt.yticks([])
plt.show()
InĀ [59]:
mnist.train.labels[5]
Out[59]:
InĀ [60]:
np.argmax(mnist.train.labels[5])
Out[60]:
Batch maker embedded
InĀ [61]:
x, y = mnist.train.next_batch(3)
print(x.shape)
print(y.shape)
- Our network model
- Network training (learning) $$\omega:= \omega - \alpha \nabla_{\omega} \left( h_{\omega} \left(x^{(i)}\right),y^{(i)}\right)$$
6.2.1. Import LibraryĀ¶
InĀ [62]:
# Import Library
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
6.2.2. Load MNIST DataĀ¶
- Download MNIST data from tensorflow tutorial example
InĀ [63]:
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("MNIST_data/", one_hot=True)
InĀ [64]:
train_x, train_y = mnist.train.next_batch(1)
img = train_x[0,:].reshape(28,28)
plt.figure(figsize=(6,6))
plt.imshow(img,'gray')
plt.title("Label : {}".format(np.argmax(train_y[0,:])))
plt.xticks([])
plt.yticks([])
plt.show()
One hot encoding
InĀ [65]:
print ('Train labels : {}'.format(train_y[0, :]))
InĀ [68]:
n_input = ?
n_hidden = ?
n_output = ?
6.2.4. Define Weights, Biases, and PlaceholderĀ¶
- Define parameters based on predefined layer size
- Initialize with normal distribution with $\mu = 0$ and $\sigma = 0.1$
InĀ [69]:
weights = {
'hidden' : ?,
'output' : ?
}
biases = {
'hidden' : ?,
'output' : ?
}
InĀ [70]:
x = tf.placeholder(?)
y = tf.placeholder(?)
6.2.5. Build a ModelĀ¶
First, the layer performs several matrix multiplication to produce a set of linear activations
Second, each linear activation is running through a nonlinear activation function
Third, predict values with an affine transformation
InĀ [71]:
# Define Network
def build_model(x, weights, biases):
# first hidden layer
hidden = tf.add(tf.matmul(x, weights['hidden']), biases['hidden'])
# non-linear activate function
hidden = tf.nn.relu(hidden)
# Output layer
output = tf.add(tf.matmul(hidden, weights['output']), biases['output'])
return output
6.2.6. Define Loss and OptimizerĀ¶
Loss
- This defines how we measure how accurate the model is during training. As was covered in lecture, during training we want to minimize this function, which will "steer" the model in the right direction.
- Classification: Cross entropy
- Equivalent to apply logistic regression
- Equivalent to apply logistic regression
Optimizer
- This defines how the model is updated based on the data it sees and its loss function.
- AdamOptimizer: the most popular optimizer
InĀ [72]:
# Define Loss
pred = build_model(x, weights, biases)
loss = tf.nn.softmax_cross_entropy_with_logits(logits = pred, labels = y)
loss = tf.reduce_mean(loss)
LR = 0.0001
optm = tf.train.AdamOptimizer(LR).minimize(loss)
6.2.7. Define Optimization Configuration and Then OptimizeĀ¶
- Define parameters for training ANN
n_batch: batch size for mini-batch gradient descentn_iter: the number of iteration stepsn_prt: check loss for everyn_prtiteration
- Metrics
- Here we can define metrics used to monitor the training and testing steps. In this example, we'll look at the accuracy, the fraction of the images that are correctly classified.
Initializer
- Initialize all the variables
InĀ [73]:
n_batch = 50 # Batch Size
n_iter = 3000 # Learning Iteration
n_prt = 250 # Print Cycle
InĀ [74]:
# open session
# define init
# run init
?
?
?
# c1 is for train
# c2 is for test
loss_record_train = []
loss_record_test = []
for epoch in range(n_iter):
train_x, train_y = mnist.train.next_batch(n_batch)
sess.run(optm, feed_dict = {x: train_x, y: train_y})
if epoch % n_prt == 0:
test_x, test_y = mnist.test.next_batch(n_batch)
c1 = sess.run(loss, feed_dict = {x: train_x, y: train_y})
c2 = sess.run(loss, feed_dict = {x: ?, y: ?})
loss_record_train.append(c1)
loss_record_test.append(c2)
print ("Iter : {}".format(epoch))
print ("Cost : {}".format(c1))
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record_train))*n_prt,
loss_record_train, label = 'training')
plt.plot(np.arange(len(loss_record_test))*n_prt,
loss_record_test, label = 'testing')
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.legend(fontsize = 12)
plt.ylim([0, np.max(loss_record_train)])
plt.show()
6.2.8. Test or EvaluateĀ¶
InĀ [75]:
test_x, test_y = mnist.test.next_batch(100)
my_pred = sess.run(pred, feed_dict = {x : test_x})
my_pred = np.argmax(my_pred, axis = 1)
labels = np.argmax(test_y, axis = 1)
accr = np.mean(np.equal(my_pred, labels))
print("Accuracy : {}%".format(accr*100))
InĀ [78]:
test_x, test_y = mnist.test.next_batch(1)
logits = sess.run(tf.nn.softmax(pred), feed_dict = {x : test_x})
predict = np.argmax(logits)
plt.figure(figsize = (6,6))
plt.imshow(test_x.reshape(28,28), 'gray')
plt.xticks([])
plt.yticks([])
plt.show()
print('Prediction : {}'.format(predict))
np.set_printoptions(precision = 2, suppress = True)
print('Probability : {}'.format(logits.ravel()))
You may observe that the accuracy on the test dataset is a little lower than the accuracy on the training dataset. This gap between training accuracy and test accuracy is an example of overfitting, when a machine learning model performs worse on new data than on its training data.
What is the highest accuracy you can achieve with this first fully connected model? Since the handwritten digit classification task is pretty straightforward, you may be wondering how we can do better...
$\Rightarrow$ As we saw in lecture, convolutional neural networks (CNNs) are particularly well-suited for a variety of tasks in computer vision, and have achieved near-perfect accuracies on the MNIST dataset. We will build a CNN and ultimately output a probability distribution over the 10 digit classes (0-9) in the next lectures.
7. AutoencoderĀ¶
7.1. Unsupervised LearningĀ¶
Definition
- Unsupervised learning refers to most attempts to extract information from a distribution that do not require human labor to annotate example
- Main task is to find the 'best' representation of the data
Dimension Reduction
- Attempt to compress as much information as possible in a smaller representation
- Preserve as much information as possible while obeying some constraint aimed at keeping the representation simpler
7.2. AutoencodersĀ¶
It is like 'deep learning version' of unsupervised learning.
Definition
- An autoencoder is a neural network that is trained to attempt to copy its input to its output
- The network consists of two parts: an encoder and a decoder that produce a reconstruction
Encoder and Decoder
- Encoder function : $z = f(x)$
- Decoder function : $x = g(z)$
- We learn to set $g\left(f(x)\right) = x$
- Autoencoder combines an encoder $f$ from the original space $\mathscr{X}$ to a latent space $\mathscr{F}$, and a decoder $g$ to map back to $\mathscr{X}$, such that $f \circ g$ is [close to] the identity on the data
$$ \mathbb{E} \left[ \lVert X - g \circ f(X) \rVert^2 \right] \approx 0$$
- A proper autoencoder has to capture a "good" parametrization of the signal, and in particular the statistical dependencies between the signal components.
7.3.1. Import LibraryĀ¶
InĀ [Ā ]:
# import os
# os.environ["CUDA_DEVICE_ORDER"]="PCI_BUS_ID"
# os.environ["CUDA_VISIBLE_DEVICES"]="0"
InĀ [79]:
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
%matplotlib inline
7.3.2. Load MNIST DataĀ¶
InĀ [80]:
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("MNIST_data/", one_hot=True)
- Only use (1, 5, 6) digits to visualize latent space in 2-D
InĀ [81]:
train_idx = ((np.argmax(mnist.train.labels, 1) == 1) | \
(np.argmax(mnist.train.labels, 1) == 5) | \
(np.argmax(mnist.train.labels, 1) == 6))
test_idx = ((np.argmax(mnist.test.labels, 1) == 1) | \
(np.argmax(mnist.test.labels, 1) == 5) | \
(np.argmax(mnist.test.labels, 1) == 6))
train_imgs = mnist.train.images[train_idx]
train_labels = mnist.train.labels[train_idx]
test_imgs = mnist.test.images[test_idx]
test_labels = mnist.test.labels[test_idx]
n_train = train_imgs.shape[0]
n_test = test_imgs.shape[0]
print ("The number of training images : {}, shape : {}".format(n_train, train_imgs.shape))
print ("The number of testing images : {}, shape : {}".format(n_test, test_imgs.shape))
7.3.3. Define a Structure of an AutoencoderĀ¶
- Input shape and latent variable shape
- Encoder shape
- Decoder shape
InĀ [82]:
# Shape of input and latent variable
n_input = ?
# Encoder structure
n_encoder1 = ?
n_encoder2 = ?
n_latent = ?
# Decoder structure
n_decoder2 = ?
n_decoder1 = ?
7.3.4. Define Weights, Biases, and PlaceholderĀ¶
- Define weights and biases for encoder and decoder, separately
- Based on the pre-defined layer size
- Initialize with normal distribution of $\mu=0$ and $\sigma=0.1$
InĀ [83]:
weights = {
'encoder1' : tf.Variable(tf.random_normal([n_input, n_encoder1], stddev = 0.1)),
'encoder2' : tf.Variable(tf.random_normal([n_encoder1, n_encoder2], stddev = 0.1)),
'latent' : tf.Variable(tf.random_normal([n_encoder2, n_latent], stddev = 0.1)),
'decoder2' : tf.Variable(tf.random_normal([n_latent, n_decoder2], stddev = 0.1)),
'decoder1' : tf.Variable(tf.random_normal([n_decoder2, n_decoder1], stddev = 0.1)),
'reconst' : tf.Variable(tf.random_normal([n_decoder1, n_input], stddev = 0.1))
}
biases = {
'encoder1' : tf.Variable(tf.random_normal([?], stddev = 0.1)),
'encoder2' : tf.Variable(tf.random_normal([?], stddev = 0.1)),
'latent' : tf.Variable(tf.random_normal([?], stddev = 0.1)),
'decoder2' : tf.Variable(tf.random_normal([?], stddev = 0.1)),
'decoder1' : tf.Variable(tf.random_normal([?], stddev = 0.1)),
'reconst' : tf.Variable(tf.random_normal([?], stddev = 0.1))
}
InĀ [84]:
x = tf.placeholder(tf.float32, [None, n_input])
7.3.5. Build a ModelĀ¶
Encoder
- Simple ANN (MLP) model
- Use
tanhfor a nonlinear activation function latentis not applied with a nonlinear activation function
Decoder
- Simple ANN (MLP) model
- Use
tanhfor a nonlinear activation function reconstis not applied with a nonlinear activation function
InĀ [85]:
def encoder(x, weights, biases):
encoder1 = tf.add(tf.matmul(x, weights['encoder1']), biases['encoder1'])
encoder1 = tf.nn.tanh(encoder1)
encoder2 = tf.add(tf.matmul(encoder1, weights['encoder2']), biases['encoder2'])
encoder2 = tf.nn.tanh(encoder2)
latent = tf.add(tf.matmul(encoder2, weights['latent']), biases['latent'])
return latent
InĀ [86]:
def decoder(latent, weights, biases):
decoder2 = tf.add(tf.matmul(latent, weights['decoder2']), biases['decoder2'])
decoder2 = tf.nn.tanh(decoder2)
decoder1 = tf.add(tf.matmul(decoder2, weights['decoder1']), biases['decoder1'])
decoder1 = tf.nn.tanh(decoder1)
reconst = tf.add(tf.matmul(decoder1, weights['reconst']), biases['reconst'])
return reconst
7.3.6. Define Loss and OptimizerĀ¶
Loss
- Squared loss
Optimizer
- AdamOptimizer: the most popular optimizer
InĀ [87]:
LR = 0.0001
latent = encoder(x, weights, biases)
reconst = decoder(latent, weights, biases)
loss = tf.square(tf.subtract(x, reconst))
loss = tf.reduce_mean(loss)
optm = tf.train.AdamOptimizer(LR).minimize(loss)
7.3.7. Define Optimization Configuration and Then OptimizeĀ¶
- Define parameters for training autoencoder
n_batch: batch size for mini-batch gradient descentn_iter: the number of iteration stepsn_prt: check loss for everyn_prtiteration
InĀ [88]:
n_batch = 50
n_iter = 2500
n_prt = 250
InĀ [89]:
def train_batch_maker(batch_size):
random_idx = np.random.randint(n_train, size = batch_size)
return train_imgs[random_idx], train_labels[random_idx]
InĀ [90]:
def test_batch_maker(batch_size):
random_idx = np.random.randint(n_test, size = batch_size)
return test_imgs[random_idx], test_labels[random_idx]
InĀ [91]:
sess = tf.Session()
init = tf.global_variables_initializer()
sess.run(init)
loss_record_train = []
loss_record_test = []
for epoch in range(n_iter):
train_x, _ = train_batch_maker(n_batch)
sess.run(optm, feed_dict = {x : train_x})
if epoch % n_prt == 0:
test_x, _ = test_batch_maker(n_batch)
c1 = sess.run(loss, feed_dict = {x: train_x})
c2 = sess.run(loss, feed_dict = {x: test_x})
loss_record_train.append(c1)
loss_record_test.append(c2)
print ("Iter : {}".format(epoch))
print ("Cost : {}".format(c1))
plt.figure(figsize=(10,8))
plt.plot(np.arange(len(loss_record_train))*n_prt, loss_record_train, label = 'training')
plt.plot(np.arange(len(loss_record_test))*n_prt, loss_record_test, label = 'testing')
plt.xlabel('iteration', fontsize = 15)
plt.ylabel('loss', fontsize = 15)
plt.legend(fontsize = 12)
plt.ylim([0,np.max(loss_record_train)])
plt.show()
7.3.8. Test or EvaluateĀ¶
- Test reconstruction performance of the autoencoder
InĀ [92]:
test_x, _ = test_batch_maker(1)
x_reconst = sess.run(reconst, feed_dict = {x: test_x})
plt.figure(figsize = (10,8))
plt.subplot(1,2,1)
plt.imshow(test_x.reshape(28,28), 'gray')
plt.title('Input Image', fontsize = 15)
plt.xticks([])
plt.yticks([])
plt.subplot(1,2,2)
plt.imshow(x_reconst.reshape(28,28), 'gray')
plt.title('Reconstructed Image', fontsize = 15)
plt.xticks([])
plt.yticks([])
plt.show()
- To see the distribution of latent variables, we make a projection of 784-dimensional image space onto 2-dimensional latent space
InĀ [93]:
test_x, test_y = test_batch_maker(500)
test_y = np.argmax(test_y, axis = 1)
test_latent = sess.run(latent, feed_dict = {x: test_x})
plt.figure(figsize = (10,10))
plt.scatter(test_latent[test_y == 1,0], test_latent[test_y == 1,1], label = '1')
plt.scatter(test_latent[test_y == 5,0], test_latent[test_y == 5,1], label = '5')
plt.scatter(test_latent[test_y == 6,0], test_latent[test_y == 6,1], label = '6')
plt.title('Latent Space', fontsize=15)
plt.xlabel('Z1', fontsize=15)
plt.ylabel('Z2', fontsize=15)
plt.legend(fontsize = 15)
plt.axis('equal')
plt.show()
Data Generation
It generates something that makes sense.
These results are unsatisfying, because the density model used on the latent space ā± is too simple and inadequate.
Building a āgoodā model amounts to our original problem of modeling an empirical distribution, although it may now be in a lower dimension space.
This is a motivation to VAE or GAN.
InĀ [94]:
new_data = np.array([[-4, 0]])
latent_input = tf.placeholder(tf.float32, [None, n_latent])
reconst = decoder(latent_input, weights, biases)
fake_image = sess.run(reconst, feed_dict = {latent_input: new_data})
plt.figure(figsize=(16,7))
plt.subplot(1,2,1)
plt.scatter(test_latent[test_y == 1,0], test_latent[test_y == 1,1], label = '1')
plt.scatter(test_latent[test_y == 5,0], test_latent[test_y == 5,1], label = '5')
plt.scatter(test_latent[test_y == 6,0], test_latent[test_y == 6,1], label = '6')
plt.scatter(new_data[:,0], new_data[:,1], c = 'k', marker = 'o', s = 200, label = 'new data')
plt.title('Latent Space', fontsize = 15)
plt.xlabel('Z1', fontsize = 15)
plt.ylabel('Z2', fontsize = 15)
plt.legend(loc = 2, fontsize = 12)
plt.axis('equal')
plt.subplot(1,2,2)
plt.imshow(fake_image.reshape(28,28), 'gray')
plt.title('Generated Fake Image', fontsize = 15)
plt.xticks([])
plt.yticks([])
plt.show()
7.4. VisualizationĀ¶
Image Generation
- Select an arbitrary latent varibale $z$
- Generate images using the learned decoder
InĀ [95]:
# Initialize canvas
nx = 20
ny = 20
x_values = np.linspace(-8, 4, nx)
y_values = np.linspace(-4, 6, ny)
canvas = np.empty((28*ny, 28*nx))
# Define placeholder
latent_input = tf.placeholder(tf.float32, [None, n_latent])
reconst = decoder(latent_input, weights, biases)
for i, yi in enumerate(y_values):
for j, xi in enumerate(x_values):
latent_ = np.array([[xi, yi]])
reconst_ = sess.run(reconst, feed_dict = {latent_input: latent_})
canvas[(nx-i-1)*28:(nx-i)*28,j*28:(j+1)*28] = reconst_.reshape(28, 28)
plt.figure(figsize = (16, 7))
plt.subplot(1,2,1)
plt.scatter(test_latent[test_y == 1,0], test_latent[test_y == 1,1], label = '1')
plt.scatter(test_latent[test_y == 5,0], test_latent[test_y == 5,1], label = '5')
plt.scatter(test_latent[test_y == 6,0], test_latent[test_y == 6,1], label = '6')
plt.title('Latent Space', fontsize = 15)
plt.xlabel('Z1', fontsize = 15)
plt.ylabel('Z2', fontsize = 15)
plt.legend(fontsize = 12)
plt.axis('equal')
plt.subplot(1,2,2)
plt.imshow(canvas, 'gray')
plt.title('Manifold', fontsize = 15)
plt.xlabel('Z1', fontsize = 15)
plt.ylabel('Z2', fontsize = 15)
plt.xticks([])
plt.yticks([])
plt.show()
7.5. Latent RepresentationĀ¶
To get an intuition of the latent representation, we can pick two samples š„ and š„ā² at random and interpolate samples along the line in the latent space
$$g((1-\alpha)f(x) + \alpha f(x'))$$
- Interpolation in High Dimension
- Interpolation in Manifold
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